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Theorem 2spthonot 25283
 Description: The set of simple paths of length 2 between two vertices (in a graph) as ordered triple. (Contributed by Alexander van der Vekens, 1-Mar-2018.)
Assertion
Ref Expression
2spthonot 2SPathOnOt SPathOn
Distinct variable groups:   ,,,   ,,,   ,,,   ,,,
Allowed substitution hints:   (,,)   (,,)

Proof of Theorem 2spthonot
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 is2spthonot 25281 . . . 4 2SPathOnOt SPathOn
21adantr 463 . . 3 2SPathOnOt SPathOn
32oveqd 6295 . 2 2SPathOnOt SPathOn
4 simprl 756 . . 3
5 simprr 758 . . 3
6 3xpexg 6585 . . . . 5
76ad2antrr 724 . . . 4
8 rabexg 4544 . . . 4 SPathOn
97, 8syl 17 . . 3 SPathOn
10 oveq12 6287 . . . . . . . 8 SPathOn SPathOn
1110breqd 4406 . . . . . . 7 SPathOn SPathOn
12 eqeq2 2417 . . . . . . . . 9
1312adantr 463 . . . . . . . 8
14 eqeq2 2417 . . . . . . . . 9
1514adantl 464 . . . . . . . 8
1613, 153anbi13d 1303 . . . . . . 7
1711, 163anbi13d 1303 . . . . . 6 SPathOn SPathOn
18172exbidv 1737 . . . . 5 SPathOn SPathOn
1918rabbidv 3051 . . . 4 SPathOn SPathOn
20 eqid 2402 . . . 4 SPathOn SPathOn
2119, 20ovmpt2ga 6413 . . 3 SPathOn SPathOn SPathOn
224, 5, 9, 21syl3anc 1230 . 2 SPathOn SPathOn
233, 22eqtrd 2443 1 2SPathOnOt SPathOn
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 367   w3a 974   wceq 1405  wex 1633   wcel 1842  crab 2758  cvv 3059   class class class wbr 4395   cxp 4821  cfv 5569  (class class class)co 6278   cmpt2 6280  c1st 6782  c2nd 6783  c1 9523  c2 10626  chash 12452   SPathOn cspthon 24922   2SPathOnOt c2pthonot 25274 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-1st 6784  df-2nd 6785  df-2spthonot 25277 This theorem is referenced by:  el2spthonot  25287  el2spthonot0  25288  2spthonot3v  25293  2pthwlkonot  25302  2spotfi  25309  2spotdisj  25478
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